Raanan Schul Yale University. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.1/53

A Characterization of Subsets of Rectifiable Curves in Hilbert Space Raanan Schul Yale University A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.1/53

Motivation Want to discuss the geometry of sets of points in Hilbert space. Results about sets lying in R d usually have constants that depend exponentially on d. This is called the curse of dimensionality A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.2/53

Outline Introduction dimension free estimates in harmonic analysis traveling salesmen theorems. Jones and Okikiolu dictionary main result - thesis work Our proof of thm 1 3 types of balls type 2 balls - more details. two subtypes type 1 (3) balls - more details A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.3/53

Dimension free estimates - a sample Theorem: ([SS83]) Ball maximal function is L p bounded. Suppose f : R d C. Let M d (f)(x) = sup r>0 1 Volume(Ball(0, r)) Ball(0,r) f(x y) dy. Then M d (f) p C p f p for 1 < p. The constant C p is independent of d. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.5/53

Dimension free estimates - a sample Theorem: ([Ste83]) Size of Riesz vector is L p bounded. If f : R d C and R j (f) is the j th Riesz transform, then ( d j=1 R j (f) 2 ) 1 2 p C p f p for 1 < p <. The constant C p is independent of d. Reminder: j th Riesz transform is defined by (R j (f))(ξ) = i ξ j ξ f(ξ) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.6/53

GMT Before quantitative rectifiabilty: A set K is called 1-rectifiable iff K i N Γ i except for a set of 1-dim Hausdorff measure 0, where Γ i is the image of a Lipschitz function (= a curve). Note that there is no mention of the length of a curve or how many of them there are! When is a set K contained inside a single curve of finite length? How long is the shortest curve? A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.8/53

Quantitative Rectifiability This was answered in the setting of R d by Peter Jones and Kate Okikiolu. Intuitive Picture: A connected set of finite length is flat on most scales and in most locations. This can be used to characterize subsets of finite length connected sets. One can give a quantitative version of this using multiresolutional analysis. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.9/53

Quantitative Rectifiability Definition: (Jones β number) β K0 (Q 0 ) = 1 diam(q 0 ) inf L line sup dist(x, L) x K 0 Q 0 = radius of the thinest tube containing K 0 Q 0 diam(q 0 ). K 0 Q 0 A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.10/53

Quantitative Rectifiability Jones ([Jon90]): Theorem 1: For any connected Γ C Q dyadic grid βγ 2 (3Q)diam(Q) l(γ) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.11/53

Quantitative Rectifiability Jones ([Jon90]): Theorem 1: For any connected Γ C Q dyadic grid βγ 2 (3Q)diam(Q) l(γ) Theorem 2: For any set K R d, there exists Γ 0 K, Γ 0 connected, such that l(γ 0 ) βk 2 (3Q)diam(Q) + diam(k) Q dyadic grid (and in particular K C). A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.11/53

Quantitative Rectifiability Jones ([Jon90]): Theorem 1: For any connected Γ C Q dyadic grid βγ 2 (3Q)diam(Q) l(γ) Theorem 2: For any set K R d, there exists Γ 0 K, Γ 0 connected, such that l(γ 0 ) βk 2 (3Q)diam(Q) + diam(k) Q dyadic grid A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.11/53

Quantitative Rectifiability Jones ([Jon90]): + Okikiolu ([Oki92]) Theorem 1: For any connected Γ C or Γ R d Q dyadic grid βγ 2 (3Q)diam(Q) l(γ) Theorem 2: For any set K R d, there exists Γ 0 K, Γ 0 connected, such that l(γ 0 ) βk 2 (3Q)diam(Q) + diam(k) Q dyadic grid A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.11/53

Corollary: For any connected set Γ R d diam(γ) + Q dyadic grid βγ 2 (3Q)diam(Q) l(γ) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.12/53

More generally: For any set K R d diam(k) + Q dyadic grid β 2 K (3Q)diam(Q) l(γ MST ) where Γ MST is the shortest curve containing K. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.13/53

Proof of corollary: l(γ MST ) l(γ 0 ) diam(k) + diam(k) + l(γ MST ) Q dyadic grid Q dyadic grid β 2 K (3Q)diam(Q) β 2 Γ MST (3Q)diam(Q) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.14/53

Allowed people to prove results about one dimensional sets in C or R d. (e.g. [BJ90]) Another example:([bj97]) Γ connected, with β Γ (3Q) ɛ, Q Γ, diam(q) diam(γ) then dim H (Γ) 1 + cɛ 2. Set the basis for The Theory of Quantitative Rectifiability (developed by David, Semmes, Pajot, Verdera, Lerman and others) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.15/53

Dictionary discovered by Peter Jones. wavelets Jones β numbers {a j,k } for function f {β(q)} for set K analysis and synthesis of the function f f 2 = a j,k 2 Wavelet square function W ψ (x) 2 Slide by Gilad Lerman. analysis and synthesis of curve Γ K l(γ) β(q)2 diam(q) + diam(γ) Jones function J (x) (define J(x) on board) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.17/53

Reminder Theorem 1: For any connected Γ R d Q dyadic grid βγ 2 (3Q)diam(Q) l(γ) Theorem 2: For any set K R d, there exists Γ 0 K, Γ 0 connected, such that l(γ 0 ) βk 2 (3Q)diam(Q) + diam(k) Q dyadic grid A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.18/53

Issue to fix Constants that make inequalities true are exponential in d. Want: Hilbert Space version ( = dim free) of above (would allow Quantitative Rectifiability in Hilbert space). Some results for Ahlfors-David curves have proofs that work (either as is or with small variations) in Hilbert space! Examples appear in [Dav91, DS93]. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.19/53

Different formulation Take {X n } n a sequence of nested nets. X n K a 2 n net, X n X n+1 G K = {B(x, A2 n ) : x X n ; n Z} Theorem 1: For any connected Γ R d, Γ K (Q)diam(Q) l(γ) Q G K β 2 Γ Theorem 2: For any set K R d, there exists Γ 0 K, Γ 0 connected, such that l(γ 0 ) (Q)diam(Q) + diam(k) Q G K β 2 K constants that make inequalities true are exponential in d A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.20/53

My Thesis Work Take {X n } n a sequence of nested nets. X n K a 2 n net, X n X n+1 G K = {B(x, A2 n ) : x X n ; n Z} Theorem 1 : For any connected Γ R d, Γ K (Q)diam(Q) l(γ) Q G K β 2 Γ Theorem 2 : For any set K R d, there exists Γ 0 K, Γ 0 connected, such that l(γ 0 ) (Q)diam(Q) + diam(k) Q G K β 2 K With constants independent of dimension! We actually show the theorems for Γ or K in Hilbert space. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.22/53

My Thesis Work Corollary: For any set K Hilbert Space diam(k) + Q G K β 2 K (Q)diam(Q) l(γ MST ) where Γ MST is the shortest curve containing K. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.23/53

Outline Introduction dimension free estimates in harmonic analysis traveling salesmen theorems. Jones and Okikiolu dictionary main result - thesis work Our proof of thm 1 (write thm 1 on board) 3 types of balls type 2 balls - more details. two subtypes type 1 (3) balls - more details A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.24/53

Pf of thm 1; (main ideas) A Few Notes Before We Start: We must deal with balls, not dyadic cubes. For each scale they are highly overlapping. It suffices to prove the theorem for Γ R d as long as the constants we get are independent of d. Given a compact connected set Γ, we fix a parametrization γ. The parametrization arclength is to the one dimensional Hausdorff length. Okikiolu s outline A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.25/53

Pf of thm 1; (main ideas) Λ(Q) := {τ Γ Q : τ Q; τ is a subarc of γ} = γ(connected components of γ 1 (Q)). (draw on board) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.26/53

Pf of thm 1; (main ideas) We will call τ : [a, b] R d Λ(Q) almost straight iff β(τ) := sup t [a,b] dist(τ(t), < [τ(a), τ(b)] >) diam(q) < ɛ 2 β(q) where τ Λ(Q) and < [x, y] > is the line containing the segment [x, y] (this is how we define the Jones β number of an arc). If < [τ(a), τ(b)] > is not well defined (i.e. τ(a) = τ(b)) take the supremum over all lines through τ(b). A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.27/53

Pf of thm 1; (main ideas) S Q := {τ Λ(Q) : β(τ) < ɛ 2 β(q)} A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.28/53

Pf of thm 1; (main ideas) S Q := {τ Λ(Q) : β(τ) < ɛ 2 β(q)} Example of Q A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.28/53

Pf of thm 1; (main ideas) S Q := {τ Λ(Q) : β(τ) < ɛ 2 β(q)} Example of S Q A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.28/53

Pf of thm 1; (main ideas) Fix γ Q Λ Q an arc containing the center of Q. Type 1: γ Q / S Q γ Q The ideas are a combination of ones that appear in [Dav91] and of Okikiolu s use of l 2 techniques. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.30/53

Pf of thm 1; (main ideas) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.31/53

Pf of thm 1; (main ideas) Type 2: γ Q S Q and ɛ 1 β(q) β SQ (Q) γ Q A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.31/53

Pf of thm 1; (main ideas) Type 2: γ Q S Q and ɛ 1 β(q) β SQ (Q) Geometric ideas. For each ball Q a density w Q is chosen, such that (small lie) supp(w Q ) Γ Q β(q)diam(q) Q x Q w Q w Q (x) 1 for a.e. x Also uses some type 1 techniques (to get rid of lie). A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.31/53

Pf of thm 1; (main ideas) Type 3: γ Q S Q and ɛ 1 β(q) β SQ (Q) γ Q Similar ideas to type1 balls. A little more technical. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.32/53

Type 2 Balls For type 2 ball Q we have a core U Q. The cores are divided into families, such that in each family have nice nesting properties. U Q Q A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.34/53

Type 2 Balls Consider one family. Have two subtypes of balls: 1 st subtype: Cβ SQ (U Q ) > β SQ (Q) 2 nd subtype: the rest τ Q U Q U Q subtype 1 subtype 2 A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.35/53

Type 2 Balls Focus on 1 st subtype such that β(u Q ) > 1 2, Q Assume Γ Q = union of straight lines Q. Lemma: Suppose type 2 balls such that β(u Q ) > 1 2, Q. Then β(q)diam(q) l(γ). Q idea : construct weights w Q supported on U Q Γ such that β(q)diam(q) Q x Q w Q w Q (x) 1 for a.e. x A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.36/53

Type 2 Balls (draw on board) U Q = ( i U Q i) R Q U Q i maximal in U Q, such that Q i R Q = U Q U Q i. i (how would we change this if we had 32 1 β(u Q) < 16 1??) A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.37/53

Type 2 Balls - Constructing w Q Set I Q := large connected component of γ Q U Q. Set U Q w Q = l(i Q ). We use U Q = ( i U Q i) R Q to construct w Q as a martingale. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.38/53

Type 2 Balls - Constructing w Q We divide the mass non-evenly : U w Q = Q w Q l(r Q ) and R Q s U Q i w Q = U Q w Q s l(i Q i) where s = l(r Q ) + i l(i Q i). We reiterate: U Q = ( U Q j) R Q then R Q w Q = U w Q Q s l(r Q ) and U Q j w Q = U w Q Q s l(i Q j) where s = l(r Q ) + j l(i Q j). A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.39/53

Type 2 Balls - Constructing w Q 2 crossing segments Figure 1: An (unnatural) example of weight distribution arising form the martingale. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.40/53

Type 2 Balls - Constructing w Q 2 crossing segments with U Q1 U Q2 U Q3. We have M = 1. We assume these are the only U Q s. Figure 1: An (unnatural) example of weight distribution arising form the martingale. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.40/53

Type 2 Balls - Constructing w Q 2 crossing segments with U Q1 U Q2 U Q3. We have M = 1. We assume these are the only U Q s. w Q1 Figure 1: An (unnatural) example of weight distribution arising form the martingale. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.40/53

Type 2 Balls - Constructing w Q 2 crossing segments with U Q1 U Q2 U Q3. We have M = 1. We assume these are the only U Q s. w Q2 Figure 1: An (unnatural) example of weight distribution arising form the martingale. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.40/53

Type 2 Balls - Constructing w Q 2 crossing segments with U Q1 U Q2 U Q3. We have M = 1. We assume these are the only U Q s. w Q3 Figure 1: An (unnatural) example of weight distribution arising form the martingale. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.40/53

Type 2 Balls - Constructing w Q 2 crossing segments with U Q1 U Q2 U Q3. We have M = 1. w Q1 + w Q2 + w Q3. We assume these are the only U Q s. Figure 1: An (unnatural) example of weight distribution arising form the martingale. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.40/53

Type 2 Balls - Constructing w Q Figure 2: A more complicated (unnatural) example of weight distribution arising form the martingale. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.41/53

Type 2 Balls Focus on 1 st subtype such that 2 M β(u Q ) < 2 M+1, Q Lemma: Suppose type 2 balls such that 2 M β(u Q ) < 2 M+1, Q. Then β(q)diam(q) M2 M l(γ). Q idea : weights w Q supported on U Q Γ such that β(q)diam(q) Q x Q w Q w Q (x) M2 M for a.e. x A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.42/53

Type 1 (and 3) Balls 0 2π A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.44/53

Type 1 (and 3) Balls Hilbert Space A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.45/53

Type 1 (and 3) Balls Lemma: Suppose we are given a family of arcs F = with the following properties: F i i=0 (1) τ F n+1 =!τ F n such that τ τ (2) τ F n = 2 nj diam(τ) A2 nj+2 (3) τ, τ F n = (τ τ ) 0, 1, 2 (the intersection is an empty set, a single point, or two points) (4) F 0 τ = F n τ n (we will call such a family a filtration). Then we have: β(τ) 2 diam(τ) l( τ) F 0 τ F proof copied almost word for word from [Oki92] A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.46/53

Type 1 (and 3) Balls How do we build a filtration relevant to our connected set? How do we relate β(q) to β(τ) for some τ in our filtration? A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.47/53

Type 1 (and 3) Balls Lemma: Given a family of arcs F 0 = properties: i=0 (1) τ Fn 0 = {τ Fn 0 : τ τ } C (2) τ Fn 0 = 2 n diam(τ) A2 n+1. F 0 i with the following Then we have 2CJ filtrations. Further more: τ F 0 n τ F nj for one of the filtrations, such that τ τ and diam(τ) < 2diam(τ ) (and hence β(τ) 1 4 β(τ )). This mapping can be made to be injective. J is a constant we fix; J 10 will be more then enough. How do we get such a family? A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.48/53

Type 1 (and 3) Balls Type 1: {γ Q } Q type 1 {τ Q } Q type 1 lemma 2 lemma 1 where τ Q γ Q chosen to have some properties. Type 3: More technical A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.49/53

Metric spaces A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.50/53

Metric spaces Use Menger curvature to define β (As shown on tuesday). We need more axioms. Didn t have enough time to get a decent list that I have enough confidence with to present. In particular, axioms to give sensible β(τ) Axioms to assure that if we have two almost straight arcs then one has a large subarc that is far away(relative to their joint β) from the other. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.51/53

Conclusion Have a characterization of subsets of rectifiable curves in Hilbert space. Have constructed weights that contain information about a large collection of straight lines that lie in Euclidean space. ideas should cary over to metric spaces with Menger curvature satisfying some axioms A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.52/53

References [BJ90] [BJ97] Christopher J. Bishop and Peter W. Jones. Harmonic measure and arclength. Ann. of Math. (2), 132(3):511 547, 1990. Christopher J. Bishop and Peter W. Jones. Wiggly sets and limit sets. Ark. Mat., 35(2):201 224, 1997. [Dav91] Guy David. Wavelets and singular integrals on curves and surfaces, volume 1465 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1991. [DS93] [Jon90] [Oki92] Guy David and Stephen Semmes. Analysis of and on uniformly rectifiable sets, volume 38 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 1993. Peter W. Jones. Rectifiable sets and the traveling salesman problem. Invent. Math., 102(1):1 15, 1990. Kate Okikiolu. Characterization of subsets of rectifiable curves in R n. J. London Math. Soc. (2), 46(2):336 348, 1992. [SS83] E. M. Stein and J.-O. Strömberg. Behavior of maximal functions in R n for large n. Ark. Mat., 21(2):259 269, 1983. [Ste83] E. M. Stein. Some results in harmonic analysis in R n, for n. Bull. Amer. Math. Soc. (N.S.), 9(1):71 73, 1983. A Characterization of Subsets of Rectifiable Curves in Hilbert Space p.53/53